Efficient Algorithms for Attributed Graph Alignment With Vanishing Edge Correlation
Ziao Wang, Weina Wang, Lele Wang · IEEE Transactions on Information Theory · 2025
Graph alignment refers to the task of finding the vertex correspondence between two correlated graphs ofnvertices. Extensive study has been done on polynomial-time algorithms for the graph alignment problem under the Erdős–Rényi graph pair model, where the two graphs are Erdős–Rényi graphs with edge probability$q_{\mathrm {u}}$, correlated under certain vertex correspondence. To achieve exact recovery of the correspondence, all existing algorithms at least require the edge correlation coefficient$\rho _{\mathrm {u}}$between the two graphs to benon-vanishingas$n\rightarrow \infty $. Moreover, it is conjectured that no polynomial-time algorithm can achieve exact recovery under vanishing edge correlation${\rho _{\mathrm {u}}}\lt 1/\mathrm {polylog}(n)$. In this paper, we show that with a vanishing amount of additionalattribute information, exact recovery is polynomial-time feasible undervanishingedge correlation${\rho _{\mathrm {u}}}\ge n^{-\Theta (1)}$. We identify alocaltree structure, which incorporates one layer of user information and one layer of attribute information, and apply the subgraph counting technique to such structures. A polynomial-time algorithm is proposed that recovers the vertex correspondence for most of the vertices, and then refines the output to achieve exact recovery. The consideration of attribute information is motivated by real-world applications like LinkedIn and Twitter, where user attributes like birthplace and education background can aid alignment.